School of Mathematics and Statistics, Wuhan University, Wuhan, People’s Republic of China Email: [email protected] Abstract: The security of a password authentication and update scheme based on elliptic curve cryptography proposed by Islam et al. [S.K. Hafizul Islam, G.P. Biswas, Design of improved password authentication and update scheme based on elliptic curve cryptography, Mathematical and C...

We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15.

We prove a general identity for a 3F2 hypergeometric function over a finite field Fq, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen’s Theorem expressing a 3F2 as the square of a 2F1. As another application, we evaluate an infinite family of 3F2(z) over Fq at z = −1/8. T...

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We discuss microscopic origin of integrability in Seiberg-Witten theory, following mostly the results of [1], as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbation in the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformizat...

We present a lower bound for the exponent of the group of rational points of an elliptic curve over a finite field. Earlier results considered finite fields Fqm where either q is fixed or m = 1 and q is prime. Here, we let both q and m vary; our estimate is explicit and does not depend on the elliptic curve.

In this paper, we examine the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. We prove that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.

The first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. The second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups. The third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part.